Question

Precise definitions for left- and right-sided limits Use the following definitions. Assume \(f\) exists for all \(x\) near a with \(x>a\). We say that the limit of \(f(x)\) as \(x\) approaches a from the right of a is \(L\) and write \(\lim _{x \rightarrow a^{+}} f(x)=L,\) iffor any \(\varepsilon> 0 \) there exists \(\delta > 0\) such that $$|f(x)-L| < \varepsilon \text { whenever } 0 < x- a < \delta $$ Assume \(f\) exists for all \(x\) near a with \(x < a .\) We say that the limit of \(f(x)\) as \(x\) approaches a from the left of a is \(L\) and write \(\lim _{x \rightarrow a^{-}} f(x)=L,\) iffor any \(\varepsilon > 0 \) there exists \(\delta > 0 \) such that $$ |f(x)-L| < \varepsilon \text { whenever } 0 < a - x < \delta $$ One-sided limit proofs Prove the following limits for $$ f(x)=\left\\{\begin{array}{ll} 3 x-4 & \text { if } x<0 \\ 2 x-4 & \text { if } x \geq 0 \end{array}\right. $$ a. \(\lim _{x \rightarrow 0^{+}} f(x)=-4\) b. \(\lim _{x \rightarrow 0^{-}} f(x)=-4 \quad\) c. \(\lim _{x \rightarrow 0} f(x)=-4\)

Short answer

Question: Prove that the one-sided limits of the given piecewise-defined function \(f(x)\) are as follows: a) \(\lim_{x \rightarrow 0^{+}} f(x) = -4\) b) \(\lim_{x \rightarrow 0^{-}} f(x) = -4\) c) \(\lim_{x \rightarrow 0} f(x) = -4\) Function: $$ f(x)=\left\{\begin{array}{ll} 3 x-4 & \text { if } x<0 \\ 2 x-4 & \text { if } x \geq 0 \end{array}\right. $$ Answer: Through a rigorous analysis and application of the precise definitions of left-sided and right-sided limits, we were able to prove the following: a) The right-sided limit of the function is \(-4\): \(\lim_{x \rightarrow 0^{+}} f(x) = -4\) b) The left-sided limit of the function is \(-4\): \(\lim_{x \rightarrow 0^{-}} f(x) = -4\) c) Since both the left and right-sided limits are equal and both are \(-4\), the overall limit exists and is equal to \(-4\): \(\lim_{x \rightarrow 0} f(x) = -4\)

Step-by-step solution

Understand the given function

First, observe the given function \(f(x)\) defined as: $$ f(x)=\left\{\begin{array}{ll} 3 x-4 & \text { if } x<0 \\ 2 x-4 & \text { if } x \geq 0 \end{array}\right. $$ This function behaves differently depending on the value of \(x\). If \(x < 0\), the function follows \(3x-4\), whereas if \(x \geq 0\), the function follows \(2x-4\).

Prove the right-sided limit (a)

We need to prove the right-sided limit: $$ \lim _{x \rightarrow 0^{+}} f(x)=-4 $$ Using the definition of the right-sided limit, for any \(\varepsilon > 0\), there exists \(\delta > 0\) such that: $$ |f(x) - (-4)| < \varepsilon \text{ whenever } 0 < x - 0 < \delta $$ Since \(x > 0\) (positive side of the limit), the function follows \(2x-4\). So, we can rewrite the inequality as: $$ |2x - (-4)| < \varepsilon \text{ whenever } 0 < x < \delta $$ Simplifying, we have: $$ |2x + 4| < \varepsilon \text{ whenever } 0 < x < \delta $$ Now, choose \(\delta = \frac{\varepsilon}{2}\). If \(0 < x < \delta\), then $$ |2x + 4| = 2|x + 2| \le 2|x| < 2\delta = \varepsilon $$ Thus, the right-sided limit is proven as: $$ \lim_{x \rightarrow 0^{+}} f(x) = -4 $$

Prove the left-sided limit (b)

Now, we need to prove the left-sided limit: $$ \lim _{x \rightarrow 0^{-}} f(x)=-4 $$ Using the definition of the left-sided limit, for any \(\varepsilon > 0\), there exists \(\delta > 0\) such that: $$ |f(x) - (-4)| < \varepsilon \text{ whenever } 0 < 0 - x < \delta $$ Since \(x < 0\) (negative side of the limit), the function follows \(3x-4\). So, the inequality can be rewritten as: $$ |3x - (-4)| < \varepsilon \text{ whenever } 0 < -x < \delta $$ Simplifying, we have: $$ |3x + 4| < \varepsilon \text{ whenever } 0 < -x < \delta $$ Now, choose \(\delta = \frac{\varepsilon}{3}\). If \(0 < -x < \delta\), then $$ |3x + 4| = 3|-x + \frac{4}{3}| \le 3|-x| < 3\delta = \varepsilon $$ Thus, the left-sided limit is proven as: $$ \lim_{x \rightarrow 0^{-}} f(x) = -4 $$

Prove the overall limit (c)

Finally, to prove the overall limit, we can state that since both left and right-sided limits are equal and both are equal to \(-4\), then the limit exists and is equal to \(-4\): $$ \lim_{x \rightarrow 0} f(x) = -4 $$

Key concepts

Limits and Continuity

Understanding limits and continuity is crucial for analyzing how functions behave, especially as inputs approach a specific value. The concept of a limit helps us comprehend the value that a function approaches as the input gets closer to a given point, even if the function doesn't actually reach that value at the point.

Continuity, on the other hand, involves limits inherently. A function is continuous at a point if it meets three conditions: it is defined at that point, the limit exists at that point, and the limit at that point equals the function's value. In essence, there are no abrupt changes, jumps, or holes in the graph of a function at the point of continuity.

When dealing with piecewise functions, as in the provided exercise, establishing limits and continuity may require checking each piece of the function separately. This ensures that the behavior of the function is fully captured as it transitions from one rule to another depending on the input value.

A key aspect of evaluating limits for continuity involves considering the one-sided limits to ensure they match at the point of interest. If both one-sided limits exist and are equal, the function can be continuous at that point, providing there are no other disturbances in the function's graph.

Limit Notation

Limit notation is the formal way of expressing the limit of a function as the input variable approaches a certain value. The notation \( \lim_{x \rightarrow a} f(x) = L \) states that as \(x\) gets closer to \(a\), the function \(f(x)\) approaches the value \(L\).

One-sided limits use a superscript '+' or '-' next to the point to indicate direction: \(\lim _{x \rightarrow a^{+}}\) for the right-sided limit, and \(\lim _{x \rightarrow a^{-}}\) for the left-sided limit. These notations help to clarify from which direction \(x\) is approaching the value of \(a\), which can reveal different behaviors in a function, particularly when dealing with piecewise functions or functions that are not continuous at the point \(a\).

In the provided exercise, separate considerations for \(x < 0\) and \(x \geq 0\) were necessary due to different expressions defining the function in each interval, which is characteristic of piecewise functions.

Epsilon-Delta Definition

The epsilon-delta definition is the formal definition of a limit and is essential for providing a rigorous foundation for calculus. According to this definition, the statement \( \lim_{x \rightarrow a} f(x) = L \) means that for every \( \varepsilon > 0 \), no matter how small, there exists a \( \delta > 0 \) such that \( |f(x) - L| < \varepsilon \) whenever \( 0 < |x - a| < \delta \).

This mathematical formalism encapsulates the idea that the function's values \(f(x)\) can be made arbitrarily close to \(L\) by requiring that \(x\) is sufficiently close to \(a\) but not equal to \(a\). In the solution provided for the exercise, we see this definition in action: the choice of \(\delta\) (which depends on \(\varepsilon\)) ensures that the function's behavior within this narrow range meets the required closeness to \(L\).

Understanding and applying the epsilon-delta definition is key to proving limits formally and was used precisely in steps 2 and 3 of the solution to confirm the one-sided limits for the given piecewise function.

Piecewise Functions

Piecewise functions are functions defined by multiple sub-functions, each of which applies to a certain interval within the domain. These functions often have different rules or expressions that govern their behavior depending on the input value.

In relation to limits, piecewise functions can present unique challenges. The limit as \(x\) approaches a particular value may differ depending on the direction of approach. This characteristic makes calculating one-sided limits particularly relevant, as it was in the given exercise where the function had different expressions for \(x<0\) and \(x\geq 0\).

To evaluate limits for piecewise functions, each 'piece' of the function must be considered where the different rules apply. Recognizing that the function's behavior can change at these boundaries helps when assessing continuity and determining whether the function smoothly transitions from one interval to the next.

By carefully considering the conditions under which each part of a piecewise function applies, one can use the standard limit processes to evaluate the behavior of the function at boundaries, which was crucial in establishing that the limit of \(f(x)\) as \(x\) approached 0 was \(-4\), regardless of the direction from which \(x\) approached.